To measure \(\dim \mathcal {V}_{m,r}\) , Kileel et al. [1]} build connections between deep polynomial networks and tensor decompositions. Specifically, polynomial networks with \(h=2\)  are connected to CP tensor decompositions [2]}, and deep polynomial networks are connected to an iterated tensor decomposition [3]}.
Based on decompositions, they prove that, for any fixed \(m\) , there exists \(\tilde{r} = r(m)\)  such that for any \(r > \tilde{r}\) , \(\dim \mathcal {V}_{m,r}\)  is bounded by
\(\nonumber \dim \mathcal {V}_{m,r}\le \min \left(m_h+\sum _{i=1}^{h}(m_{i-1}-1)m_i, m_h\binom{m_0+r^{h-1}+1}{r^{h-1}} \right).\) 
